Subgroups — C₆₇×#20107 #30699358

Order 17152.

598 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×SD₁₆×C₆₇	C₂	1	2	yes	maximal, complemented
2	C₆₇×#3329	C₂	1	2	yes	maximal
3	C₆₇×#1716	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#1691	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1684	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1651	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1634	C₂	1	1	yes	maximal, characteristic
8	C₆₇×#1633	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#1588	C₂	1	2	yes	maximal, complemented
10	Q₈²×C₆₇	C₂²	1	1	yes	characteristic
11	#555×C₆₇	C₂²	1	1	yes	characteristic
12	D₄×Q₈×C₆₇	C₂²	1	2	yes	—
13	#498×C₆₇	—	2	2	no	—
14	#498×C₆₇	C₂²	1	2	yes	—
15	#498×C₆₇	C₂²	1	1	yes	characteristic
16	#494×C₆₇	C₂²	1	1	yes	characteristic
17	#487×C₆₇	C₂²	1	2	yes	—
18	#485×C₆₇	—	2	2	no	—
19	#477×C₆₇	C₂²	1	2	yes	—
20	#476×C₆₇	—	2	2	no	—
21	#476×C₆₇	C₂²	1	2	yes	—
22	#475×C₆₇	—	2	2	no	—
23	#475×C₆₇	C₂²	1	2	yes	—
24	#468×C₆₇	—	2	2	no	—
25	#461×C₆₇	C₂²	1	2	yes	—
26	#448×C₆₇	—	2	2	no	—
27	C₈×Q₈×C₆₇	C₂²	1	2	yes	—
28	C₄×SD₁₆×C₆₇	—	2	2	no	—
29	C₄×SD₁₆×C₆₇	C₂²	1	2	yes	—
30	#436×C₆₇	C₂²	1	2	yes	—
31	#419×C₆₇	—	2	2	no	—
32	#335×C₆₇	C₂²	1	2	yes	—
33	#333×C₆₇	C₂²	1	1	yes	characteristic
34	#333×C₆₇	C₂²	1	1	yes	characteristic
35	#332×C₆₇	C₂²	1	2	yes	—
36	Q₈⋊C₈×C₆₇	C₂²	1	2	yes	—
37	D₄⋊C₈×C₆₇	C₂²	1	2	yes	—
38	#322×C₆₇	C₂²	1	1	yes	characteristic
39	C₂²×Q₈×C₆₇	—	2	2	no	—
40	C₂×Q₁₆×C₆₇	—	4	1	no	—
41	C₂×SD₁₆×C₆₇	—	2	2	no	—
42	C₂×SD₁₆×C₆₇	D₄	1	2	yes	—
43	C₂×M₁₆×C₆₇	—	2	2	no	—
44	C₄⋊Q₈×C₆₇	—	2	2	no	—
45	C₄⋊Q₈×C₆₇	—	2	2	no	—
46	C₄⋊Q₈×C₆₇	—	4	1	no	—
47	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
48	C₄⋊Q₈×C₆₇	C₂³	1	1	yes	characteristic
49	C₄⋊Q₈×C₆₇	C₂³	1	1	yes	characteristic
50	#124×C₆₇	—	4	1	no	—
51	C₂²⋊Q₈×C₆₇	—	2	2	no	—
52	C₂²⋊Q₈×C₆₇	—	4	2	no	—
53	#121×C₆₇	—	2	2	no	—
54	C₄×Q₈×C₆₇	—	2	2	no	—
55	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
56	C₄×D₄×C₆₇	—	2	2	no	—
57	C₄×D₄×C₆₇	C₂³	1	2	yes	—
58	#117×C₆₇	—	2	2	no	—
59	#107×C₆₇	—	2	2	no	—
60	#106×C₆₇	—	2	2	no	—
61	#106×C₆₇	—	2	2	no	—
62	#106×C₆₇	—	2	2	no	—
63	#106×C₆₇	—	2	2	no	—
64	#106×C₆₇	—	4	1	no	—
65	#106×C₆₇	D₄	1	2	yes	—
66	C₄⋊C₈×C₆₇	—	2	2	no	—
67	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
68	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
69	#103×C₆₇	—	2	2	no	—
70	#103×C₆₇	—	2	2	no	—
71	#103×C₆₇	—	2	2	no	—
72	#103×C₆₇	—	2	2	no	—
73	#103×C₆₇	—	2	2	no	—
74	#103×C₆₇	D₄	1	2	yes	—
75	#102×C₆₇	—	2	2	no	—
76	#102×C₆₇	D₄	1	2	yes	—
77	C₂²⋊C₈×C₆₇	—	2	2	no	—
78	C₄×C₈×C₆₇	—	2	2	no	—
79	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
80	C₄×C₈×C₆₇	C₂³	1	2	yes	—
81	C₂×Q₈×C₆₇	—	2	2	no	—
82	C₂×Q₈×C₆₇	—	2	1	no	—
83	C₂×Q₈×C₆₇	—	2	1	no	—
84	C₂×Q₈×C₆₇	—	4	2	no	—
85	C₂×Q₈×C₆₇	—	4	1	no	—
86	C₂×Q₈×C₆₇	—	4	4	no	—
87	C₂×Q₈×C₆₇	—	8	2	no	—
88	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
89	C₂×D₄×C₆₇	—	4	1	no	—
90	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
91	C₂²×C₄×C₆₇	—	2	2	no	—
92	C₂²×C₄×C₆₇	—	4	2	no	—
93	Q₁₆×C₆₇	—	8	2	no	—
94	SD₁₆×C₆₇	—	2	4	no	—
95	SD₁₆×C₆₇	—	8	2	no	—
96	M₁₆×C₆₇	—	8	2	no	—
97	C₂×C₈×C₆₇	—	2	1	no	—
98	C₂×C₈×C₆₇	—	2	2	no	—
99	C₂×C₈×C₆₇	—	2	2	no	—
100	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
101	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
102	C₄⋊C₄×C₆₇	—	2	2	no	—
103	C₄⋊C₄×C₆₇	—	2	2	no	—
104	C₄⋊C₄×C₆₇	—	2	2	no	—
105	C₄⋊C₄×C₆₇	—	2	2	no	—
106	C₄⋊C₄×C₆₇	—	2	2	no	—
107	C₄⋊C₄×C₆₇	—	4	2	no	—
108	C₄⋊C₄×C₆₇	—	4	2	no	—
109	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
110	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
111	C₂²⋊C₄×C₆₇	—	2	2	no	—
112	C₂²⋊C₄×C₆₇	—	4	2	no	—
113	C₂²⋊C₄×C₆₇	—	4	2	no	—
114	C₄²×C₆₇	—	2	2	no	—
115	C₄²×C₆₇	—	4	1	no	—
116	C₄²×C₆₇	—	4	1	no	—
117	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
118	C₂³×C₆₇	—	2	2	no	—
119	Q₈×C₆₇	—	2	2	no	—
120	Q₈×C₆₇	—	4	2	no	—
121	Q₈×C₆₇	—	4	2	no	—
122	Q₈×C₆₇	—	8	4	no	—
123	Q₈×C₆₇	—	8	1	no	—
124	Q₈×C₆₇	—	8	1	no	—
125	Q₈×C₆₇	—	16	1	no	—
126	Q₈×C₆₇	#137	1	4	yes	—
127	D₄×C₆₇	—	2	2	no	—
128	D₄×C₆₇	—	2	2	no	—
129	D₄×C₆₇	—	16	1	no	—
130	C₂×C₄×C₆₇	—	2	2	no	—
131	C₂×C₄×C₆₇	—	2	2	no	—
132	C₂×C₄×C₆₇	—	2	2	no	—
133	C₂×C₄×C₆₇	—	4	2	no	—
134	C₂×C₄×C₆₇	—	4	2	no	—
135	C₂×C₄×C₆₇	—	4	1	no	—
136	C₂×C₄×C₆₇	—	4	1	no	—
137	C₂×C₄×C₆₇	—	4	2	no	—
138	C₂×C₄×C₆₇	—	8	4	no	—
139	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
140	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
141	C₈×C₆₇	—	2	2	no	—
142	C₈×C₆₇	—	2	2	no	—
143	C₈×C₆₇	—	2	2	no	—
144	C₈×C₆₇	—	4	2	no	—
145	C₈×C₆₇	—	4	2	no	—
146	C₂²×C₆₇	—	4	2	no	—
147	C₂²×C₆₇	—	4	2	no	—
148	C₂²×C₆₇	—	4	2	no	—
149	C₂²×C₆₇	D₄²	1	1	yes	characteristic
150	C₄×C₆₇	—	2	4	no	—
151	C₄×C₆₇	—	2	1	no	—
152	C₄×C₆₇	—	4	2	no	—
153	C₄×C₆₇	—	4	2	no	—
154	C₄×C₆₇	—	8	2	no	—
155	C₄×C₆₇	—	8	1	no	—
156	C₄×C₆₇	—	8	1	no	—
157	C₄×C₆₇	C₂×#137	1	2	yes	—
158	C₄×C₆₇	#577	1	2	yes	—
159	#20107	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
160	C₂×C₆₇	—	8	2	no	—
161	C₂×C₆₇	#3266	1	2	yes	—
162	C₂×C₆₇	#3291	1	1	yes	characteristic
163	Q₈×SD₁₆	C₂×C₆₇	1	2	yes	complemented
164	#3329	C₂×C₆₇	1	2	yes	—
165	#1716	C₂×C₆₇	1	1	yes	characteristic, complemented
166	#1691	C₂×C₆₇	1	2	yes	complemented
167	#1684	C₂×C₆₇	1	2	yes	complemented
168	#1651	C₂×C₆₇	1	2	yes	complemented
169	#1634	C₂×C₆₇	1	1	yes	characteristic
170	#1633	C₂×C₆₇	1	1	yes	characteristic, complemented
171	#1588	C₂×C₆₇	1	2	yes	complemented
172	C₆₇	#20107	1	1	yes	characteristic, direct product factor, Sylow
173	Q₈²	C₂²×C₆₇	1	1	yes	characteristic
174	#555	C₂²×C₆₇	1	1	yes	characteristic
175	D₄×Q₈	C₂²×C₆₇	1	2	yes	—
176	#498	—	2	2	no	—
177	#498	C₂²×C₆₇	1	2	yes	—
178	#498	C₂²×C₆₇	1	1	yes	characteristic
179	#494	C₂²×C₆₇	1	1	yes	characteristic
180	#487	C₂²×C₆₇	1	2	yes	—
181	#485	—	2	2	no	—
182	#477	C₂²×C₆₇	1	2	yes	—
183	#476	—	2	2	no	—
184	#476	C₂²×C₆₇	1	2	yes	—
185	#475	—	2	2	no	—
186	#475	C₂²×C₆₇	1	2	yes	—
187	#468	—	2	2	no	—
188	#461	C₂²×C₆₇	1	2	yes	—
189	#448	—	2	2	no	—
190	C₈×Q₈	C₂²×C₆₇	1	2	yes	—
191	C₄×SD₁₆	—	2	2	no	—
192	C₄×SD₁₆	C₂²×C₆₇	1	2	yes	—
193	#436	C₂²×C₆₇	1	2	yes	—
194	#419	—	2	2	no	—
195	#335	C₂²×C₆₇	1	2	yes	—
196	#333	C₂²×C₆₇	1	1	yes	characteristic
197	#333	C₂²×C₆₇	1	1	yes	characteristic
198	#332	C₂²×C₆₇	1	2	yes	—
199	Q₈⋊C₈	C₂²×C₆₇	1	2	yes	—
200	D₄⋊C₈	C₂²×C₆₇	1	2	yes	—
201	#322	C₂²×C₆₇	1	1	yes	characteristic
202	C₂²×Q₈	—	2	2	no	—
203	C₂×Q₁₆	—	4	1	no	—
204	C₂×SD₁₆	—	2	2	no	—
205	C₂×SD₁₆	D₄×C₆₇	1	2	yes	—
206	C₂×M₁₆	—	2	2	no	—
207	C₄⋊Q₈	—	2	2	no	—
208	C₄⋊Q₈	—	2	2	no	—
209	C₄⋊Q₈	—	4	1	no	—
210	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
211	C₄⋊Q₈	C₂³×C₆₇	1	1	yes	characteristic
212	C₄⋊Q₈	C₂³×C₆₇	1	1	yes	characteristic
213	#124	—	4	1	no	—
214	C₂²⋊Q₈	—	2	2	no	—
215	C₂²⋊Q₈	—	4	2	no	—
216	#121	—	2	2	no	—
217	C₄×Q₈	—	2	2	no	—
218	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
219	C₄×D₄	—	2	2	no	—
220	C₄×D₄	C₂³×C₆₇	1	2	yes	—
221	#117	—	2	2	no	—
222	#107	—	2	2	no	—
223	#106	—	2	2	no	—
224	#106	—	2	2	no	—
225	#106	—	2	2	no	—
226	#106	—	2	2	no	—
227	#106	—	4	1	no	—
228	#106	D₄×C₆₇	1	2	yes	—
229	C₄⋊C₈	—	2	2	no	—
230	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
231	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
232	#103	—	2	2	no	—
233	#103	—	2	2	no	—
234	#103	—	2	2	no	—
235	#103	—	2	2	no	—
236	#103	—	2	2	no	—
237	#103	D₄×C₆₇	1	2	yes	—
238	#102	—	2	2	no	—
239	#102	D₄×C₆₇	1	2	yes	—
240	C₂²⋊C₈	—	2	2	no	—
241	C₄×C₈	—	2	2	no	—
242	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
243	C₄×C₈	C₂³×C₆₇	1	2	yes	—
244	C₂×Q₈	—	2	2	no	—
245	C₂×Q₈	—	2	1	no	—
246	C₂×Q₈	—	2	1	no	—
247	C₂×Q₈	—	4	2	no	—
248	C₂×Q₈	—	4	1	no	—
249	C₂×Q₈	—	4	4	no	—
250	C₂×Q₈	—	8	2	no	—
251	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
252	C₂×D₄	—	4	1	no	—
253	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
254	C₂²×C₄	—	2	2	no	—
255	C₂²×C₄	—	4	2	no	—
256	Q₁₆	—	8	2	no	—
257	SD₁₆	—	2	4	no	—
258	SD₁₆	—	8	2	no	—
259	M₁₆	—	8	2	no	—
260	C₂×C₈	—	2	1	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
264	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	2	2	no	—
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	4	2	no	—
271	C₄⋊C₄	—	4	2	no	—
272	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
273	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
274	C₂²⋊C₄	—	2	2	no	—
275	C₂²⋊C₄	—	4	2	no	—
276	C₂²⋊C₄	—	4	2	no	—
277	C₄²	—	2	2	no	—
278	C₄²	—	4	1	no	—
279	C₄²	—	4	1	no	—
280	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
281	C₂³	—	2	2	no	—
282	Q₈	—	2	2	no	—
283	Q₈	—	4	2	no	—
284	Q₈	—	4	2	no	—
285	Q₈	—	8	4	no	—
286	Q₈	—	8	1	no	—
287	Q₈	—	8	1	no	—
288	Q₈	—	16	1	no	—
289	Q₈	#137×C₆₇	1	4	yes	—
290	D₄	—	2	2	no	—
291	D₄	—	2	2	no	—
292	D₄	—	16	1	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	—	4	1	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	8	4	no	—
302	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
303	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
304	C₈	—	2	2	no	—
305	C₈	—	2	2	no	—
306	C₈	—	2	2	no	—
307	C₈	—	4	2	no	—
308	C₈	—	4	2	no	—
309	C₂²	—	4	2	no	—
310	C₂²	—	4	2	no	—
311	C₂²	—	4	2	no	—
312	C₂²	D₄²×C₆₇	1	1	yes	characteristic
313	C₄	—	2	4	no	—
314	C₄	—	2	1	no	—
315	C₄	—	4	2	no	—
316	C₄	—	4	2	no	—
317	C₄	—	8	2	no	—
318	C₄	—	8	1	no	—
319	C₄	—	8	1	no	—
320	C₄	C₂×#137×C₆₇	1	2	yes	—
321	C₄	#577×C₆₇	1	2	yes	—
322	C₂	—	8	2	no	—
323	C₂	C₆₇×#3266	1	2	yes	—
324	C₂	C₆₇×#3291	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₆₇
prime radical	#555×C₆₇	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₄²×C₆₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. Q₈×SD₁₆×C₆₇	C₂	semidirect
3. C₆₇×#1716	C₂	semidirect
4. C₆₇×#1691	C₂	semidirect
5. C₆₇×#1684	C₂	semidirect
6. C₆₇×#1651	C₂	semidirect
8. C₆₇×#1633	C₂	semidirect
9. C₆₇×#1588	C₂	semidirect
159. #20107	C₆₇	direct (N × H)
163. Q₈×SD₁₆	C₂×C₆₇	semidirect
165. #1716	C₂×C₆₇	semidirect
166. #1691	C₂×C₆₇	semidirect
167. #1684	C₂×C₆₇	semidirect
168. #1651	C₂×C₆₇	semidirect
170. #1633	C₂×C₆₇	semidirect
171. #1588	C₂×C₆₇	semidirect
172. C₆₇	#20107	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
Q₈×SD₁₆×C₆₇	2
C₆₇×#3329	2
C₆₇×#1716	2
C₆₇×#1691	2
C₆₇×#1684	2
C₆₇×#1651	2
C₆₇×#1634	2
C₆₇×#1633	2
C₆₇×#1588	2
#20107	67

Subgroups — C67×#20107 #30699358

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#20107 #30699358